I have an $n\times 1$ vector $x$ and an $n\times n$ diagonal matrix $\Lambda$. The diagonal entries of $\Lambda$ are all unique and nonzero.
I am interested in the quadratic form associated with $\Lambda$. Formally, the function $f(x): \mathbb{R}^n \rightarrow \mathbb{R}$ with $$f(x) = x^T \Lambda x$$
Specifically, if I restrict $x$ so that it lies on the surface $x^T x = \eta$ for some scalar $\eta$, can I make any statements about the quadratic form $f(x)$?